182 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



If we attempted to carry out this process rigorously for a charged sphere the 

 problem would he very complex, because, in addition to the linear motion of the 

 sphere, rotation would also be set up. Doubtless the problem is well worth 

 investigation, but it is beyond the scope of the equations so far developed. 



It appears, however, that for a wave-length of incident waves which is large in 

 comparison with the radius of the sphere, by far the most important term in the 

 incident waves is that corresponding to a spherical harmonic of the first order. This 

 is the term which gives rise to linear motion of the sphere with associated first order 

 vibrations. I therefore propose to limit the calculation to this order. 



The equations at the beginning of this section have now to be modified by the 

 introduction of the harmonic term due to the exciting waves, and we might then 

 proceed to complete determination of ^, I//! and t/i 2 . We may, however, with 

 advantage, simplify the matter at the outset by remembering that for such a high 

 value of K as we contemplate, i/^ and \// 2 are in general of order 1/K 1 s as compared 

 with y and f. 



Thus for a train of waves in the direction of z, for which the electrical force parallel 

 to x is Fe I/j(l K " ) , the equations for the first order vibrations are 



1 a ' ka 



1 , 1 / _ fj\ _ <jF ., {('] -/:V) sin ka-ka cos ka} jikct 

 X a X + a-\ X CJ~~ " C 8 " W 



These equations are exact for a conductor and approximately true for an insulator, 

 the terms neglected being of order 1/K. 



Taking the real part of the solution, we get 



1 H k 2 u 2 } sin kaka cos ka I 

 m I 



2 



, 7,2-,2 



+ A/ a 



m 



tn 



These give the forced part of the excited motion, and we should have to add terms 

 depending on the free vibrations. 



It is generally supposed that a vibratory motion of a charged sphere is attended 

 by the emission of radiation. This is proved by application of the Poynting flux, 

 and the field is supposed to be determined by a function ^, which is identified with 

 ef/C, while the exciting field is totally neglected. Now the exciting field must be 



